In a trapezoid with bases 10 and 6, the smaller diagonal is perpendicular to the bases.

| April 30, 2021 | education

In a trapezoid with bases 10 and 6, the smaller diagonal is perpendicular to the bases. The acute angles add up to 90 degrees. Find the sides.

Let us prove that triangles ABD and BCD are similar.

By condition, the sum of acute angles is 90, BCD + BAD = 90. Then ABD angle BAD = 90 – BCD.

In a triangle ВСD, angle ВDC = 90 – ВСD. Then the angle BDC of triangle BCD is equal to the angle BAD of triangle ABD, therefore, triangles ABD and BCD are similar in acute angle.

Then: AD / BD = BD / BC.

BD ^ 2 = AD * AC = 10 * 6 = 60.

Then, by the Pythagorean theorem, AB ^ 2 = AD ^ 2 + BD ^ 2 = 100 + 60 = 160.

AB = √160 = 4 * √10 cm.

CD ^ 2 = BC ^ 2 + BD ^ 2 = 36 + 60 = 96.

СD = √96 = 4 * √6 cm.

Answer: The sides of the trapezoid are equal to AB = 4 * √10 cm and CD = 4 * √6 cm.



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